Existence of quasilinear relaxation shock profiles in systems with characteristic velocities

Guy Métivier[1]; Benjamin Texier[2]; Kevin Zumbrun[3]

  • [1] IMB, Université de Bordeaux, CNRS, IMB, 33405 Talence Cedex, France
  • [2] Université Paris Diderot (Paris 7), Institut de Mathématiques de Jussieu, UMR CNRS 7586
  • [3] Indiana University, Bloomington, IN 47405

Annales de la faculté des sciences de Toulouse Mathématiques (2012)

  • Volume: 21, Issue: 1, page 1-23
  • ISSN: 0240-2963

Abstract

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We revisit the existence problem for shock profiles in quasilinear relaxation systems in the case that the velocity is a characteristic mode, implying that the profile ODE is degenerate. Our result states existence, with sharp rates of decay and distance from the Chapman–Enskog approximation, of small-amplitude quasilinear relaxation shocks. Our method of analysis follows the general approach used by Métivier and Zumbrun in the semilinear case, based on Chapman–Enskog expansion and the macro–micro decomposition of Liu and Yu. In the quasilinear case, however, in order to close the analysis, we find it necessary to apply a parameter-dependent Nash-Moser iteration due to Texier and Zumbrun, whereas, in the semilinear case, a simple contraction-mapping argument sufficed.

How to cite

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Métivier, Guy, Texier, Benjamin, and Zumbrun, Kevin. "Existence of quasilinear relaxation shock profiles in systems with characteristic velocities." Annales de la faculté des sciences de Toulouse Mathématiques 21.1 (2012): 1-23. <http://eudml.org/doc/251020>.

@article{Métivier2012,
abstract = {We revisit the existence problem for shock profiles in quasilinear relaxation systems in the case that the velocity is a characteristic mode, implying that the profile ODE is degenerate. Our result states existence, with sharp rates of decay and distance from the Chapman–Enskog approximation, of small-amplitude quasilinear relaxation shocks. Our method of analysis follows the general approach used by Métivier and Zumbrun in the semilinear case, based on Chapman–Enskog expansion and the macro–micro decomposition of Liu and Yu. In the quasilinear case, however, in order to close the analysis, we find it necessary to apply a parameter-dependent Nash-Moser iteration due to Texier and Zumbrun, whereas, in the semilinear case, a simple contraction-mapping argument sufficed.},
affiliation = {IMB, Université de Bordeaux, CNRS, IMB, 33405 Talence Cedex, France; Université Paris Diderot (Paris 7), Institut de Mathématiques de Jussieu, UMR CNRS 7586; Indiana University, Bloomington, IN 47405},
author = {Métivier, Guy, Texier, Benjamin, Zumbrun, Kevin},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {sharp rates of decay; Chapman-Enskog approximation; small-amplitude; relaxation shocks; Nash-Moser iteration},
language = {eng},
month = {1},
number = {1},
pages = {1-23},
publisher = {Université Paul Sabatier, Toulouse},
title = {Existence of quasilinear relaxation shock profiles in systems with characteristic velocities},
url = {http://eudml.org/doc/251020},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Métivier, Guy
AU - Texier, Benjamin
AU - Zumbrun, Kevin
TI - Existence of quasilinear relaxation shock profiles in systems with characteristic velocities
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/1//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - 1
SP - 1
EP - 23
AB - We revisit the existence problem for shock profiles in quasilinear relaxation systems in the case that the velocity is a characteristic mode, implying that the profile ODE is degenerate. Our result states existence, with sharp rates of decay and distance from the Chapman–Enskog approximation, of small-amplitude quasilinear relaxation shocks. Our method of analysis follows the general approach used by Métivier and Zumbrun in the semilinear case, based on Chapman–Enskog expansion and the macro–micro decomposition of Liu and Yu. In the quasilinear case, however, in order to close the analysis, we find it necessary to apply a parameter-dependent Nash-Moser iteration due to Texier and Zumbrun, whereas, in the semilinear case, a simple contraction-mapping argument sufficed.
LA - eng
KW - sharp rates of decay; Chapman-Enskog approximation; small-amplitude; relaxation shocks; Nash-Moser iteration
UR - http://eudml.org/doc/251020
ER -

References

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